We are given a nice subgraph \(H\)  of \(G\) 
embedded on a 1-punctured plane \(\boxdot \) .
Then we show how to remove \(g(k)\) -isolated vertices from \(H\)  so that \(H\)  has treewidth \(2^{O(k)}\) .
Our algorithm is simple: we remove all vertices sufficiently far from the boundary vertices of \(H\)  with respect to the radial distance.
Let \(V_i\)  be the set of vertices \(v\)  of \(H\)  such that the minimum radial distance between \(v\)  and a boundary vertex of \(H\)  is exactly \(i\) .
That is, \(V_0\)  is the set of boundary vertices of \(H\) , and
\(V_{i+1}\)  is the set of vertices not in \(V_i\cup V_{i-1}\)  lying on a face incident to a vertex of \(V_i\) .
We can partition \(V(H)\)  into \(V_0,\ldots V_{n^{\prime }}\)  in this way in time linear in the complexity of \(H\) 
using a doubly connected edge list of \(H\) , which a data structure for representing an embedding of a planar graph [1]}.
